BV weak solutions to Gauss¨CCodazzi system for isometric immersions

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• 作者：Cleopatra  ; Christoforou ;   ; ^{Christoforou.Cleopatra@ucy.ac.cy}
• 关键词：Isometric immersion problem ; Gauss curvature ; First and second fundamental forms ; Systems of balance laws ; Bounded variation ; Random choice method
• 刊名：Journal of Differential Equations
• 出版年：2012
• 出版时间：1 February 2012
• 年：2012
• 卷：252
• 期：3
• 页码：2845-2863
• 全文大小：233 K

文摘

The isometric immersion problem for surfaces embedded into is studied via the fluid dynamic framework introduced in Chen et al. (2010) as a system of balance laws of mixed-type. The techniques developed in the theory of weak solutions of bounded variation in continuum physics are employed to deal with the isometric immersions in the setting of differential geometry. The so-called BV framework is formed that establishes convergence of approximate solutions of bounded variation to the Gauss¨CCodazzi system and yields the isometric realization of two-dimensional surfaces into . Local and global existence results are established for weak solutions of small bounded variation to the Gauss¨CCodazzi system for negatively curved surfaces that admit equilibrium configurations. As an application, the case of catenoidal shell of revolution is provided.